# Properties

 Label 588.4.a.a Level $588$ Weight $4$ Character orbit 588.a Self dual yes Analytic conductor $34.693$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.6931230834$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 14q^{5} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 14q^{5} + 9q^{9} + 4q^{11} - 54q^{13} + 42q^{15} + 14q^{17} - 92q^{19} - 152q^{23} + 71q^{25} - 27q^{27} - 106q^{29} + 144q^{31} - 12q^{33} + 158q^{37} + 162q^{39} + 390q^{41} - 508q^{43} - 126q^{45} + 528q^{47} - 42q^{51} + 606q^{53} - 56q^{55} + 276q^{57} + 364q^{59} - 678q^{61} + 756q^{65} + 844q^{67} + 456q^{69} - 8q^{71} + 422q^{73} - 213q^{75} + 384q^{79} + 81q^{81} + 548q^{83} - 196q^{85} + 318q^{87} - 1194q^{89} - 432q^{93} + 1288q^{95} + 1502q^{97} + 36q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −3.00000 0 −14.0000 0 0 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.a.a 1
3.b odd 2 1 1764.4.a.l 1
4.b odd 2 1 2352.4.a.v 1
7.b odd 2 1 84.4.a.b 1
7.c even 3 2 588.4.i.h 2
7.d odd 6 2 588.4.i.a 2
21.c even 2 1 252.4.a.a 1
21.g even 6 2 1764.4.k.n 2
21.h odd 6 2 1764.4.k.c 2
28.d even 2 1 336.4.a.e 1
35.c odd 2 1 2100.4.a.g 1
35.f even 4 2 2100.4.k.g 2
56.e even 2 1 1344.4.a.p 1
56.h odd 2 1 1344.4.a.b 1
84.h odd 2 1 1008.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 7.b odd 2 1
252.4.a.a 1 21.c even 2 1
336.4.a.e 1 28.d even 2 1
588.4.a.a 1 1.a even 1 1 trivial
588.4.i.a 2 7.d odd 6 2
588.4.i.h 2 7.c even 3 2
1008.4.a.d 1 84.h odd 2 1
1344.4.a.b 1 56.h odd 2 1
1344.4.a.p 1 56.e even 2 1
1764.4.a.l 1 3.b odd 2 1
1764.4.k.c 2 21.h odd 6 2
1764.4.k.n 2 21.g even 6 2
2100.4.a.g 1 35.c odd 2 1
2100.4.k.g 2 35.f even 4 2
2352.4.a.v 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 14$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(588))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$14 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$54 + T$$
$17$ $$-14 + T$$
$19$ $$92 + T$$
$23$ $$152 + T$$
$29$ $$106 + T$$
$31$ $$-144 + T$$
$37$ $$-158 + T$$
$41$ $$-390 + T$$
$43$ $$508 + T$$
$47$ $$-528 + T$$
$53$ $$-606 + T$$
$59$ $$-364 + T$$
$61$ $$678 + T$$
$67$ $$-844 + T$$
$71$ $$8 + T$$
$73$ $$-422 + T$$
$79$ $$-384 + T$$
$83$ $$-548 + T$$
$89$ $$1194 + T$$
$97$ $$-1502 + T$$
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